We are proposing the following theorems (along with proofs) (See attached Images) as a new technique for use in ascertaining convergence of non-monotone series with smooth differentiable functions. We will be using this in another important proposal later on.
I am of an Actuarial background as as such I will kindly ignore all references to my notation rather than trying to see the simple series of integrals for what they are.
We put our ideas on this platform for constructive criticism.
Dear Rajah Iyer ,
What is the relation between f(x) and the sequence of functions
{fn(x)}. Is the sequence converge to f(x)?
Next, we can discuss your claim.
Best regards
Dear Issam I will add more images to explain:
The series is Σ f(x)| 1∞ i.e. f(1) + f(2) + .... , Here f is smooth and continuous over [1, ∞) and has to have the property that f(∞) = c \in R
we are using the simple equation that f(c) = -∫c ∞ f'(x) dx , where f'(x) = d/dx f(x).
We then have that since f(1) = -∫1 ∞ f'(x) dx and f(2) = -∫2 ∞ f'(x) dx, f(1) + f(2) = -∫1 ∞ f'(x) dx - ∫2 ∞ f'(x) dx, and as such the area under [f(1) + f(2)] over [2,∞) is double that of f(2), and triple for f(3) and so on.....
Dear Rajah Iyer ,
It means that fn = f(n).
To support your claim, let we try Σ (1/n2 ), it is a convergent sequence.
We apply your test:
The integral(h goes to infinity) lim ∫h₁ (xdf/dx)= lim ∫h₁ (x)(-2)(x)-3dx
= 2lim (x-1 )∞1 = -2, exists.
Am I right?
Now, your claim that for any f(n), your test is true. Isn't it?
Best regards
Yes Dear Professor Issam you are correct. and the proof is quite elegant (if I do say so myself :)) Please do look through it. We(I) apply this to a very difficult problem in my dissertation.
Best wishes
Rajah
Dear Rajah,
Before giving any judge to your proposed test, I ask you to apply the test on the following convergent series:
∑(sin n)/(n²)
Do you think that ∫x(d(sin x/x²)/dx )dx exists?
where the boundaries of the integral are 1 and h, h goes to infinity.
Best regards
Dear Issam Kaddoura Kindly see attached. Also we can successfully apply the test alone if and only if x(d/dx f(x)) is analytically integrable. (Re Uploading Images)
Therefore 100% our test demonstrates the convergence of your series.
Dear Rajah Iyer ,
We are in process to test a claim!
So we seek a counterexample!
What about constant functions?
fn = c is constant.
Σ(c) is divergent but the integral is convergent.
Is there any explanation?
Best regards
Dear @Issam the test specifically requires the function f in Σ(f) to be convergent, i.e. a primary requirement is that Limx->{infinity} f(x) = 0. In the case of a constant this is not the case.
for years I've tried for counterexamples sir, and the proof is is also evident :)
Dear Rajah Iyer ,
(*) Dear @Issam the test specifically requires the function f in Σ(f) to be convergent, i.e. a primary requirement is that Limx->{infinity} f(x) = 0
This is a wrong response, your claim was
the sequence Σ(f)is convergent if and only if ∫x(df/dx )dx exists.
where the boundaries of the integral are 1 and h, h goes to infinity.
(*) primary requirement is that Limx->{infinity} f(x) = 0.
You didn't mention this condition in your previous posts.
You need to change the (wrong) statement in image 1.
And add this new additional condition as I mentioned in my previous answer.
(*) I've tried for counterexamples sir, and the proof is is also evident.
You didn't show any proof to judge.
This is not a good way to deal with RG followers who are trying to help.
You should confess with any shortage or missing requirements.
Do you wait for another family for counterexamples?
Then you adjust your claim ( as a new primary requirement!!).
The right procedure is to show your (FULL PROOF) to judge.
In the meanwhile, you can publish your work (if available) in the arXiv website
https://arxiv.org/archive/math
to save your copyright.!
Dear Issam Kaddoura you are completely correct I apologize, I seem (in the very first image) to have missed a portion of the sentence and this is a fault on my part (it is present in the abstract and I make use of it implicitly incorrectly so).
"This is a wrong response, your claim was
the sequence Σ(f)is convergent if and only if ∫x(df/dx )dx exists."
I kindly have attached the image displaying the proof (for which I kindly welcome all feedback) and reasoning in the first set of images when beginning the discussion. My fault again for not presenting this in a manner more complete and conducive. I have taken the images and put these into a simple document attached here. (Done in this way to remove a lot of unnecessary text.)
This article is being prepared for the ArXiv (albeit far from ready ) and many thanks for your suggestion.
Many thanks and best wishes
Rajah
Many thanks to all for the wonderful feedback.
I will be sure to amend the shortcomings in the proposal to include in addition 'continuous smooth and differentiable over [1,infinity)'
Any and all feedback on the proof would be most appreciated.
Dear Rajah Iyer , I think you result is straight forward from Euler's Summation formula.
Hi Tapas Chatterjee its a common misconception that that is the case. There are a handful of convergence establishment techniques, the closest to this is the integral test and the expression that we have come up with is not straightforward in that sense hence not made use of before, and there would not then be a need for a whole host of techniques for non-monotone series etc...
I see that you have seen:
https://www.researchgate.net/post/Application_of_new_techniques_to_the_Riemann_Zeta_Function
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